nLab display locale

Idea

The category of cosheaves of sets on a locale $X$ is equivalent to the category of complete spreads over $X$. The equivalence functor sends a cosheaf $D$ of sets over $X$ to its display locale over $X$.

Thus, the functor described here plays the same role in the equivalence between cosheaves of sets on $X$ and complete spreads over $X$ as the etale locale? construction plays in the equivalence between sheaves of sets on $X$ and etale locales? over $X$.

Definition

The display locale of a precosheaf $D$ on a locale $X$ is a locale over $X$ (i.e., an object in the slice category $Loc/X$) given by the base change of the morphism

$$Alex(elem(D))\to Alex(elem(1_X))$$

(induced by the unique morphism $D\to 1_X$) along the morphism $X\to Alex(elem(1_X))$.

Here $1_X$ denotes the terminal precosheaf on $X$.

The functor $elem\colon Precosheaf(X) \to Poset$ computes the category of elements of a precosheaf on $X$.

The functor $Alex\colon Poset \to Locale$ computes the Alexandroff locale of a poset $P$ by sending it the locale whose frame consists of downward closed subsets of $P$.

The morphism $X\to Alex(elem(1_X))$ has as its underlying morphism of frames the map that sends a downward closed subset of $X$ to its supremum.

Adjunction

The display locale functor

$$Cosheaf(X) \to LocConLoc/X$$

is right adjoint to the cosheaf of connected components functor

$$LocConLoc/X \to Cosheaf(X).$$

(The above adjunction also makes holds for precosheaves if we generalize the cosheaf of connected components construction accordingly, see Section 2 in Funk.)

This adjunction exhibits the category of cosheaves of sets on $X$ as a (full) reflective subcategory of locally connected locales over $X$. The essential image of the inclusion is known as the category of complete spreads over $X$.

References

• Jonathon Funk, The display locale of a cosheaf.